Chromatic bounds for the subclasses of $p{K}_2$-free graphs

Abstract

The concept of $\chi$-binding functions for classes of free graphs has been extensively studied in the past. In this paper, we improve the existing $\chi$-binding function for $\{2K_2,K_1+C_4\}$-free graphs. Also, we find a linear $\chi$-binding function for $\{2K_2,K_2+P_4\}$-free graphs. In addition, we give alternative proofs for the $\chi$-binding function of $\{2K_2,gem\}$-free graphs, $\{2K_2,HVN\}$-free graphs and $\{2K_2,K_5-e\}$-free graphs. Finally, for $p\ge 3$, we find polynomial $\chi$-binding functions for $\{p{K}_2,H\}$-free graphs where $H\in \{gem,diamond,K_2+P_4,HVN,K_5-e,butterfly,ge{m}^{+},dart,K_1+C_4,C_4,\overline{P_5}\}$.